Method and apparatus for modeling dynamic and steady-state processes for prediction, control and optimization

ABSTRACT

A method for providing independent static and dynamic models in a prediction, control and optimization environment utilizes an independent static model ( 20 ) and an independent dynamic model ( 22 ). The static model ( 20 ) is a rigorous predictive model that is trained over a wide range of data, whereas the dynamic model ( 22 ) is trained over a narrow range of data. The gain K of the static model ( 20 ) is utilized to scale the gain k of the dynamic model ( 22 ). The forced dynamic portion of the model ( 22 ) referred to as the b l  variables are scaled by the ratio of the gains K and k. The b i  have a direct effect on the gain of a dynamic model ( 22 ). This is facilitated by a coefficient modification block ( 40 ). Thereafter, the difference between the new value input to the static model ( 20 ) and the prior steady-state value is utilized as an input to the dynamic model ( 22 ). The predicted dynamic output is then summed with the previous steady-state value to provide a predicted value Y. Additionally, the path that is traversed between steady-state value changes.

CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is a Continuation of U.S. patent applicationSer. No. 09/250,432, filed Feb. 16, 1999, entitled “METHOD AND APPARATUSFOR MODELING DYNAMIC AND STEADY-STATE PROCESSES FOR PREDICTION, CONTROLAND OPTIMIZATION,” now U.S. Pat. No. 6,487,459, issued Nov. 26, 2002,which is a Continuation of U.S. application Ser. No. 08/643,464, filedon May 6, 1996, now U.S. Pat. No. 5,933,345, issued Aug. 3, 1999,entitled “METHOD AND APPARATUS FOR DYNAMIC AND STEADY STATE MODELINGOVER A DESIRED PATH BETWEEN TWO END POINTS.”

TECHNICAL FIELD OF THE INVENTION

[0002] The present invention pertains in general to modeling techniquesand, more particularly, to combining steady-state and dynamic models forthe purpose of prediction, control and optimization.

BACKGROUND OF THE INVENTION

[0003] Process models that are utilized for prediction, control andoptimization can be divided into two general categories, steady-statemodels and dynamic models. In each case the model is a mathematicalconstruct that characterizes the process, and process measurements areutilized to parameterize or fit the model so that it replicates thebehavior of the process. The mathematical model can then be implementedin a simulator for prediction or inverted by an optimization algorithmfor control or optimization.

[0004] Steady-state or static models are utilized in modem processcontrol systems that usually store a great deal of data, this datatypically containing steady-state information at many differentoperating conditions. The steady-state information is utilized to traina non-linear model wherein the process input variables are representedby the vector U that is processed through the model to output thedependent variable Y. The non-linear model is a steady-statephenomenological or empirical model developed utilizing several orderedpairs (U_(i), Y_(i)) of data from different measured steady states. If amodel is represented as:

Y=P(U, Y)  (001)

[0005] where P is some parameterization, then the steady-state modelingprocedure can be presented as:

({right arrow over (U)}, {right arrow over (Y)})→P  (002)

[0006] where U and Y are vectors containing the U_(i), Y_(l) orderedpair elements. Given the model P, then the steady-state process gain canbe calculated as: $\begin{matrix}{K = \frac{\Delta \quad {P\left( {U,Y} \right)}}{\Delta \quad U}} & (003)\end{matrix}$

[0007] The steady-state model therefore represents the processmeasurements that are taken when the system is in a “static” mode. Thesemeasurements do not account for the perturbations that exist whenchanging from one steady-state condition to another steady-statecondition. This is referred to as the dynamic part of a model.

[0008] A dynamic model is typically a linear model and is obtained fromprocess measurements which are not steady-state measurements; rather,these are the data obtained when the process is moved from onesteady-state condition to another steady-state condition. This procedureis where a process input or manipulated variable u(t) is input to aprocess with a process output or controlled variable y(t) being outputand measured. Again, ordered pairs of measured data (u(I), y(I)) can beutilized to parameterize a phenomenological or empirical model, thistime the data coming from non-steady-state operation. The dynamic modelis represented as:

y(t)=p(u(t), y(t))  (004)

[0009] where p is some parameterization. Then the dynamic modelingprocedure can be represented as:

({right arrow over (u)}, {right arrow over (y)})→p  (005)

[0010] Where u and y are vectors containing the (u(I),y(I)) ordered pairelements. Given the model p, then the steady-state gain of a dynamicmodel can be calculated as: $\begin{matrix}{k = \frac{\Delta \quad {p\left( {u,y} \right)}}{\Delta \quad u}} & (006)\end{matrix}$

[0011] Unfortunately, almost always the dynamic gain k does not equalthe steady-state gain K, since the steady-state gain is modeled on amuch larger set of data, whereas the dynamic gain is defined around aset of operating conditions wherein an existing set of operatingconditions are mildly perturbed. This results in a shortage ofsufficient non-linear information in the dynamic data set in whichnon-linear information is contained within the static model. Therefore,the gain of the system may not be adequately modeled for an existing setof steady-state operating conditions. Thus, when considering twoindependent models, one for the steady-state model and one for thedynamic model, there is a mis-match between the gains of the two modelswhen used for prediction, control and optimization. The reason for thismis-match are that the steady-state model is non-linear and the dynamicmodel is linear, such that the gain of the steady-state model changesdepending on the process operating point, with the gain of the linearmodel being fixed. Also, the data utilized to parameterize the dynamicmodel do not represent the complete operating range of the process,i.e., the dynamic data is only valid in a narrow region. Further, thedynamic model represents the acceleration properties of the process(like inertia) whereas the steady-state model represents the tradeoffsthat determine the process final resting value (similar to the tradeoffbetween gravity and drag that determines terminal velocity in freefall).

[0012] One technique for combining non-linear static models and lineardynamic models is referred to as the Hammerstein model. The Hammersteinmodel is basically an input-output representation that is decomposedinto two coupled parts. This utilizes a set of intermediate variablesthat are determined by the static models which are then utilized toconstruct the dynamic model. These two models are not independent andare relatively complex to create.

SUMMARY OF THE INVENTION

[0013] The present invention disclosed and claimed herein comprises amethod and apparatus for controlling the operation of a plant bypredicting a change in the dynamic input values to the plant to effect achange in the output from a current output value at a first time to adesired output value at a second time. The controller includes a dynamicpredictive model fore receiving the current input value and the desiredoutput value and predicting a plurality of input values at differenttime positions between the first time and the second time to define adynamic operation path of the plant between the current output value andthe desired output value at the second time. An optimizer then optimizesthe operation of the dynamic controller at each of the different timepositions from the first time to the second time in accordance with apredetermined optimization method that optimizes the objectives of thedynamic controller to achieve a desired path. This allows the objectivesof the dynamic predictive model to vary as a function of time.

[0014] In another aspect of the present invention, the dynamic modelincludes a dynamic forward model operable to receive input values ateach of the time positions and map the received input values through astored representation of the plant to provide a predicted dynamic outputvalue. An error generator then compares the predicted dynamic outputvalue to the desired output value and generates a primary error value asa difference therebetween for each of the time positions. An errorminimization device then determines a change in the input value tominimize the primary error value output by the error generator. Asummation device sums the determined input change value with theoriginal input value for each time position to provide a future inputvalue, with a controller controlling the operation of the errorminimization device and the optimizer. This minimizes the primary errorvalue in accordance with the predetermined optimization method.

[0015] In a yet another aspect of the present invention, the controlleris operable to control the summation device to iteratively minimize theprimary error value by storing the summed output value from thesummation device in a first pass through the error minimization deviceand then input the latch contents to the dynamic forward model insubsequent passes and for a plurality of subsequent passes. The outputof the error minimization device is then summed with the previouscontents of the latch, the latch containing the current value of theinput on the first pass through the dynamic forward model and the errorminimization device. The controller outputs the contents of the latch asthe input to the plant after the primary error value has been determinedto meet the objectives in accordance with the predetermined optimizationmethod.

[0016] In a further aspect of the present invention, a gain adjustmentdevice is provided to adjust the gain of the linear model forsubstantially all of the time positions. This gain adjustment deviceincludes a non-linear model for receiving an input value and mapping thereceived input value through a stored representation of the plant toprovide on the output thereof a predicted output value, and having anon-linear gain associated therewith. The linear model has parametersassociated therewith that define the dynamic gain thereof with aparameter adjustment device then adjusting the parameters of the linearmodel as a function of the gain of the non-linear model for at least oneof the time positions.

[0017] In yet a further aspect of the present invention, the gainadjustment device further allows for approximation of the dynamic gainfor a plurality of the time positions between the value of the dynamicgain at the first time and the determined dynamic gain at one of thetime positions having the dynamic gain thereof determined by theparameter adjustment device. This one time position is the maximum ofthe time positions at the second time.

[0018] In yet another aspect of the present invention, the errorminimization device includes a primary error modification device formodifying the primary error to provide a modified error value. The errorminimization device optimizes the operation of the dynamic controller tominimize the modified error value in accordance with the predeterminedoptimization method. The primary error is weighted as a function of timefrom the first time to the second time, with the weighting functiondecreasing as a function of time such that the primary error value isattenuated at a relatively high value proximate to the first time andattenuated at a relatively low level proximate to the second time.

[0019] In yet a further aspect of the present invention, a predictivesystem is provided for predicting the operation of a plant with thepredictive system having an input for receiving input value and anoutput for providing a predicted output value. The system includes anon-linear model having an input for receiving the input value andmapping it across a stored learned representation of the plant toprovide a predicted output. The non-linear model has an integrityassociated therewith that is a function of a training operation thatvaries across the mapped space. A first principles model is alsoprovided for providing a calculator representation of the plant. Adomain analyzer determines when the input value falls within a region ofthe mapped space having an integrity associated therewith that is lessthan a predetermined integrity threshold. A domain switching device isoperable to switch operation between the non-linear model and the firstprinciples model as a function of the determined integrity levelcomparison with the predetermined threshold. If it is above theintegrity threshold, the non-linear model is utilized and, if it isbelow the integrity threshold, the first principles model is utilized.

BRIEF DESCRIPTION OF THE DRAWINGS

[0020] For a more complete understanding of the present invention andthe advantages thereof, reference is now made to the followingdescription taken in conjunction with the accompanying Drawings inwhich:

[0021]FIG. 1 illustrates a prior art Hammerstein model;

[0022]FIG. 2 illustrates a block diagram of the modeling technique ofthe present invention;

[0023]FIGS. 3a-3 d illustrate timing diagrams for the various outputs ofthe system of FIG. 2;

[0024]FIG. 4 illustrates a detailed block diagram of the dynamic modelutilizing the identification method;

[0025]FIG. 5 illustrates a block diagram of the operation of the modelof FIG. 4;

[0026]FIG. 6 illustrates an example of the modeling technique of thepresent invention utilized in a control environment;

[0027]FIG. 7 illustrates a diagrammatic view of a change between twosteady-state values;

[0028]FIG. 8 illustrates a diagrammatic view of the approximationalgorithm for changes in the steady-state value;

[0029]FIG. 9 illustrates a block diagram of the dynamic model;

[0030]FIG. 10 illustrates a detail of the control network utilizing theerror constraining algorithm of the present invention;

[0031]FIGS. 11a and 11 b illustrate plots of the input and output duringoptimization;

[0032]FIG. 12 illustrates a plot depicting desired and predictedbehavior;

[0033]FIG. 13 illustrates various plots for controlling a system toforce the predicted behavior to the desired behavior;

[0034]FIG. 14 illustrates a plot of the trajectory weighting algorithmof the present invention;

[0035]FIG. 15 illustrates a plot for the constraining algorithm;

[0036]FIG. 16 illustrates a plot of the error algorithm as a function oftime;

[0037]FIG. 17 illustrates a flowchart depicting the statistical methodfor generating the filter and defining the end point for theconstraining algorithm of FIG. 15;

[0038]FIG. 18 illustrates a diagrammatic view of the optimizationprocess;

[0039]FIG. 18a illustrates a diagrammatic representation of the mannerin which the path between steady-state values is mapped through theinput and output space;

[0040]FIG. 19 illustrates a flowchart for the optimization procedure;

[0041]FIG. 20 illustrates a diagrammatic view of the input space and theerror associated therewith;

[0042]FIG. 21 illustrates a diagrammatic view of the confidence factorin the input space;

[0043]FIG. 22 illustrates a block diagram of the method for utilizing acombination of a non-linear system and a first principal system; and

[0044]FIG. 23 illustrates an alternate embodiment of the embodiment ofFIG. 22.

DETAILED DESCRIPTION OF THE INVENTION

[0045] Referring now to FIG. 1, there is illustrated a diagrammatic viewof a Hammerstein model of the prior art. This is comprised of anon-linear static operator model 10 and a linear dynamic model 12, bothdisposed in a series configuration. The operation of this model isdescribed in H. T. Su, and T. J. McAvoy, “Integration of MultilayerPerceptron Networks and Linear Dynamic Models: A Hammerstein ModelingApproach” to appear in I & EC Fundamentals, paper dated Jul. 7, 1992,which reference is incorporated herein by reference. Hammerstein modelsin general have been utilized in modeling non-linear systems for sometime. The structure of the Hammerstein model illustrated in FIG. 1utilizes the non-linear static operator model 10 to transform the inputU into intermediate variables H. The non-linear operator is usuallyrepresented by a finite polynomial expansion. However, this couldutilize a neural network or any type of compatible modeling system. Thelinear dynamic operator model 12 could utilize a discreet dynamictransfer function representing the dynamic relationship between theintermediate variable H and the output Y. For multiple input systems,the non-linear operator could utilize a multilayer neural network,whereas the linear operator could utilize a two layer neural network. Aneural network for the static operator is generally well known anddescribed in U.S. Pat. No. 5,353,207, issued Oct. 4, 1994, and assignedto the present assignee, which is incorporated herein by reference.These type of networks are typically referred to as a multilayerfeed-forward network which utilizes training in the form ofback-propagation. This is typically performed on a large set of trainingdata. Once trained, the network has weights associated therewith, whichare stored in a separate database.

[0046] Once the steady-state model is obtained, one can then choose theoutput vector from the hidden layer in the neural network as theintermediate variable for the Hammerstein model. In order to determinethe input for the linear dynamic operator, u(t), it is necessary toscale the output vector h(d) from the non-linear static operator model10 for the mapping of the intermediate variable h(t) to the outputvariable of the dynamic model y(t), which is determined by the lineardynamic model.

[0047] During the development of a linear dynamic model to represent thelinear dynamic operator, in the Hammerstein model, it is important thatthe steady-state non-linearity remain the same. To achieve this goal,one must train the dynamic model subject to a constraint so that thenon-linearity learned by the steady-state model remains unchanged afterthe training. This results in a dependency of the two models on eachother.

[0048] Referring now to FIG. 2, there is illustrated a block diagram ofthe modeling method of the present invention, which is referred to as asystematic modeling technique. The general concept of the systematicmodeling technique in the present invention results from the observationthat, while process gains (steady-state behavior) vary with U's andY's,(i.e., the gains are non-linear), the process dynamics seeminglyvary with time only, (i.e., they can be modeled as locally linear, buttime-varied). By utilizing non-linear models for the steady-statebehavior and linear models for the dynamic behavior, several practicaladvantages result. They are as follows:

[0049] 1. Completely rigorous models can be utilized for thesteady-state part. This provides a credible basis for economicoptimization.

[0050] 2. The linear models for the dynamic part can be updated on-line,i.e., the dynamic parameters that are known to be time-varying can beadapted slowly.

[0051] 3. The gains of the dynamic models and the gains of thesteady-state models can be forced to be consistent (k=K).

[0052] With further reference to FIG. 2, there are provided a static orsteady-state model 20 and a dynamic model 22. The static model 20, asdescribed above, is a rigorous model that is trained on a large set ofsteady-state data. The static model 20 will receive a process input Uand provide a predicted output Y. These are essentially steady-statevalues. The steady-state values at a given time are latched in variouslatches, an input latch 24 and an output latch 26. The latch 24 containsthe steady-state value of the input U_(ss), and the latch 26 containsthe steady-state output value Y_(ss). The dynamic model 22 is utilizedto predict the behavior of the plant when a change is made from asteady-state value of Y_(ss) to a new value Y. The dynamic model 22receives on the input the dynamic input value u and outputs a predicteddynamic value y. The value u is comprised of the difference between thenew value U and the steady-state value in the latch 24, U_(ss). This isderived from a subtraction circuit 30 which receives on the positiveinput thereof the output of the latch 24 and on the negative inputthereof the new value of U. This therefore represents the delta changefrom the steady-state. Similarly, on the output the predicted overalldynamic value will be the sum of the output value of the dynamic model,y, and the steady-state output value stored in the latch 26, Y_(ss).These two values are summed with a summing block 34 to provide apredicted output Y. The difference between the value output by thesumming junction 34 and the predicted value output by the static model20 is that the predicted value output by the summing junction 20accounts for the dynamic operation of the system during a change. Forexample, to process the input values that are in the input vector U bythe static model 20, the rigorous model, can take significantly moretime than running a relatively simple dynamic model. The method utilizedin the present invention is to force the gain of the dynamic model 22k_(d) to equal the gain K_(ss) of the static model 20.

[0053] In the static model 20, there is provided a storage block 36which contains the static coefficients associated with the static model20 and also the associated gain value K_(ss). Similarly, the dynamicmodel 22 has a storage area 38 that is operable to contain the dynamiccoefficients and the gain value k_(d). One of the important aspects ofthe present invention is a link block 40 that is operable to modify thecoefficients in the storage area 38 to force the value of k_(d) to beequal to the value of K_(ss). Additionally, there is an approximationblock 41 that allows approximation of the dynamic gain k_(d) between themodification updates.

[0054] Systematic Model

[0055] The linear dynamic model 22 can generally be represented by thefollowing equations: $\begin{matrix}{{\delta \quad {y(t)}} = {{\sum\limits_{i = 1}^{n}{b_{i}\delta \quad {u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta \quad {y\left( {t - i} \right)}}}}} & (007)\end{matrix}$

[0056] where:

δy(t)−y(t)−Y _(ss)  (008)

δu(t)=u(t)−u _(ss)  (009)

[0057] and t is time, a_(l) and b_(l) are real numbers, d is a timedelay, u(t) is an input and y(t) an output. The gain is represented by:$\begin{matrix}{\frac{y(B)}{u(B)} = {k = \frac{\left( {\sum\limits_{i = 1}^{n}{b_{i}B^{i - 1}}} \right)B^{d}}{1 + {\sum\limits_{i = 1}^{n}{a_{i}B^{i - 1}}}}}} & (10)\end{matrix}$

[0058] where B is the backward shift operator B(x(t))=x(t−1), t=time,the a_(i) and b_(i) are real numbers, I is the number of discreet timeintervals in the dead-time of the process, and n is the order of themodel. This is a general representation of a linear dynamic model, ascontained in George E. P. Box and G. M. Jenkins, “TIME SERIES ANALYSISforecasting and control”, Holden-Day, San Francisco, 1976, Section 10.2,Page 345. This reference is incorporated herein by reference.

[0059] The gain of this model can be calculated by setting the value ofB equal to a value of “1”. The gain will then be defined by thefollowing equation: $\begin{matrix}{\left\lbrack \frac{y(B)}{u(B)} \right\rbrack_{B = 1} = {k_{d} = \frac{\sum\limits_{i = 1}^{n}b_{i}}{1 + {\sum\limits_{i = 1}^{n}a_{i}}}}} & (11)\end{matrix}$

[0060] The a_(l) contain the dynamic signature of the process, itsunforced, natural response characteristic. They are independent of theprocess gain. The b_(l) contain part of the dynamic signature of theprocess; however, they alone contain the result of the forced response.The b_(l) determine the gain k of the dynamic model. See: J. L. Shearer,A. T. Murphy, and H. H. Richardson, “Introduction to System Dynamics”,Addison-Wesley, Reading, Mass., 1967, Chapter 12. This reference isincorporated herein by reference.

[0061] Since the gain K_(ss) of the steady-state model is known, thegain k_(d) of the dynamic model can be forced to match the gain of thesteady-state model by scaling the b_(i) parameters. The values of thestatic and dynamic gains are set equal with the value of b_(i) scaled bythe ratio of the two gains: $\begin{matrix}{\left( b_{i} \right)_{scaled} = {\left( b_{i} \right)_{old}\left( \frac{K_{ss}}{k_{d}} \right)}} & (12) \\{\left( b_{i} \right)_{scaled} = \frac{\left( b_{i} \right)_{old}{K_{ss}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (13)\end{matrix}$

[0062] This makes the dynamic model consistent with its steady-statecounterpart. Therefore, each time the steady-state value changes, thiscorresponds to a gain K_(ss) of the steady-state model. This value canthen be utilized to update the gain k_(d) of the dynamic model and,therefore, compensate for the errors associated with the dynamic modelwherein the value of k_(d) is determined based on perturbations in theplant on a given set of operating conditions. Since all operatingconditions are not modeled, the step of varying the gain will accountfor changes in the steady-state starting points.

[0063] Referring now to FIGS. 3a-3 d, there are illustrated plots of thesystem operating in response to a step function wherein the input valueU changes from a value of 100 to a value of 110. In FIG. 3a, the valueof 100 is referred to as the previous steady-state value U_(ss). In FIG.3b, the value of u varies from a value of 0 to a value of 10, thisrepresenting the delta between the steady-state value of U_(ss) to thelevel of 110, represented by reference numeral 42 in FIG. 3a. Therefore,in FIG. 3b the value of u will go from 0 at a level 44, to a value of 10at a level 46. In FIG. 3c, the output Y is represented as having asteady-state value Y_(ss) of 4 at a level 48. When the input value Urises to the level 42 with a value of 110, the output value will rise.This is a predicted value. The predicted value which is the properoutput value is represented by a level 50, which level 50 is at a valueof 5. Since the steady-state value is at a value of 4, this means thatthe dynamic system must predict a difference of a value of 1. This isrepresented by FIG. 3d wherein the dynamic output value y varies from alevel 54 having a value of 0 to a level 56 having a value of 1.0.However, without the gain scaling, the dynamic model could, by way ofexample, predict a value for y of 1.5, represented by dashed level 58,if the steady-state values were outside of the range in which thedynamic model was trained. This would correspond to a value of 5.5 at alevel 60 in the plot of FIG. 3c. It can be seen that the dynamic modelmerely predicts the behavior of the plant from a starting point to astopping point, not taking into consideration the steady-state values.It assumes that the steady-state values are those that it was trainedupon. If the gain k_(d) were not scaled, then the dynamic model wouldassume that the steady-state values at the starting point were the samethat it was trained upon. However, the gain scaling link between thesteady-state model and the dynamic model allow the gain to be scaled andthe parameter b_(l) to be scaled such that the dynamic operation isscaled and a more accurate prediction is made which accounts for thedynamic properties of the system.

[0064] Referring now to FIG. 4, there is illustrated a block diagram ofa method for determining the parameters a_(i), b_(i). This is usuallyachieved through the use of an identification algorithm, which isconventional. This utilizes the (u(t),y(t)) pairs to obtain the a_(l)and b_(i) parameters. In the preferred embodiment, a recursiveidentification method is utilized where the a_(i) and b_(i) parametersare updated with each new (u_(l)(t),y_(i)(t)) pair. See: T Eykhoff,“System Identification”, John Wiley & Sons, New York, 1974, Pages 38 and39, et. seq., and H. Kurz and W. Godecke, “Digital Parameter-AdaptiveControl Processes with Unknown Dead Time”, Automatica, Vol. 17, No. 1,1981, pp. 245-252, which references are incorporated herein byreference.

[0065] In the technique of FIG. 4, the dynamic model 22 has the outputthereof input to a parameter-adaptive control algorithm block 60 whichadjusts the parameters in the coefficient storage block 38, which alsoreceives the scaled values of k, b_(l). This is a system that is updatedon a periodic basis, as defined by timing block 62. The controlalgorithm 60 utilizes both the input u and the output y for the purposeof determining and updating the parameters in the storage area 38.

[0066] Referring now to FIG. 5, there is illustrated a block diagram ofthe preferred method. The program is initiated in a block 68 and thenproceeds to a function block 70 to update the parameters a_(i), b_(i)utilizing the (u(I),y(I)) pairs. Once these are updated, the programflows to a function block 72 wherein the steady-state gain factor K isreceived, and then to a function block 74 to set the dynamic gain to thesteady state gain, i.e., provide the scaling function describedhereinabove. This is performed after the update. This procedure can beused for on-line identification, non-linear dynamic model prediction andadaptive control.

[0067] Referring now to FIG. 6, there is illustrated a block diagram ofone application of the present invention utilizing a controlenvironment. A plant 78 is provided which receives input values u(t) andoutputs an output vector y(t). The plant 78 also has measurable statevariables s(t). A predictive model 80 is provided which receives theinput values u(t) and the state variables s(t) in addition to the outputvalue y(t). The steady-state model 80 is operable to output a predictedvalue of both y(t) and also of a future input value u(t+1). Thisconstitutes a steady-state portion of the system. The predictedsteady-state input value is U_(ss) with the predicted steady-stateoutput value being Y_(ss). In a conventional control scenario, thesteady-state model 80 would receive as an external input a desired valueof the output y^(d)(t) which is the desired value that the overallcontrol system seeks to achieve. This is achieved by controlling adistributed control system (DCS) 86 to produce a desired input to theplant. This is referred to as u(t+1), a future value. Withoutconsidering the dynamic response, the predictive model 80, asteady-state model, will provide the steady-state values. However, whena change is desired, this change will effectively be viewed as a “stepresponse”.

[0068] To facilitate the dynamic control aspect, a dynamic controller 82is provided which is operable to receive the input u(t), the outputvalue y(t) and also the steady-state values U_(ss) and Y_(ss) andgenerate the output u(t+1). The dynamic controller effectively generatesthe dynamic response between the changes, i.e., when the steady-statevalue changes from an initial steady-state value U_(ss) ^(l), Y^(l)_(ss) to a final steady-state value U^(f) _(ss), Y^(f) _(ss).

[0069] During the operation of the system, the dynamic controller 82 isoperable in accordance with the embodiment of FIG. 2 to update thedynamic parameters of the dynamic controller 82 in a block 88 with again link block 90, which utilizes the value K_(ss) from a steady-stateparameter block in order to scale the parameters utilized by the dynamiccontroller 82, again in accordance with the above described method. Inthis manner, the control function can be realized. In addition, thedynamic controller 82 has the operation thereof optimized such that thepath traveled between the initial and final steady-state values isachieved with the use of the optimizer 83 in view of optimizerconstraints in a block 85. In general, the predicted model (steady-statemodel) 80 provides a control network function that is operable topredict the future input values. Without the dynamic controller 82, thisis a conventional control network which is generally described in U.S.Pat. No. 5,353,207, issued Oct. 4, 1994, to the present assignee, whichpatent is incorporated herein by reference.

[0070] Approximate Systematic Modeling

[0071] For the modeling techniques described thus far, consistencybetween the steady-state and dynamic models is maintained by rescalingthe b_(l) parameters at each time step utilizing equation 13. If thesystematic model is to be utilized in a Model Predictive Control (MPC)algorithm, maintaining consistency may be computationally expensive.These types of algorithms are described in C. E. Garcia, D. M. Prett andM. Morari. Model predictive control: theory and practice—a survey,Automatica, 25:335-348, 1989; D. E. Seborg, T. F. Edgar, and D. A.Mellichamp. Process Dynamics and Control. John Wiley and Sons, New York,N.Y., 1989. These references are incorporated herein by reference. Forexample, if the dynamic gain k_(d) is computed from a neural networksteady-state model, it would be necessary to execute the neural networkmodule each time the model was iterated in the MPC algorithm. Due to thepotentially large number of model iterations for certain MPC problems,it could be computationally expensive to maintain a consistent model. Inthis case, it would be better to use an approximate model which does notrely on enforcing consistencies at each iteration of the model.

[0072] Referring now to FIG. 7, there is illustrated a diagram for achange between steady state values. As illustrated, the steady-statemodel will make a change from a steady-state value at a line 100 to asteady-state value at a line 102. A transition between the twosteady-state values can result in unknown settings. The only way toinsure that the settings for the dynamic model between the twosteady-state values, an initial steady-state value K_(ss) ^(l) and afinal steady-state gain K_(ss) ^(f) would be to utilize a stepoperation, wherein the dynamic gain k_(d) was adjusted at multiplepositions during the change. However, this may be computationallyexpensive. As will be described hereinbelow, an approximation algorithmis utilized for approximating the dynamic behavior between the twosteady-state values utilizing a quadratic relationship. This is definedas a behavior line 104, which is disposed between an envelope 106, whichbehavior line 104 will be described hereinbelow.

[0073] Referring now to FIG. 8, there is illustrated a diagrammatic viewof the system undergoing numerous changes in steady-state value asrepresented by a stepped line 108. The stepped line 108 is seen to varyfrom a first steady-state value at a level 110 to a value at a level 112and then down to a value at a level 114, up to a value at a level 116and then down to a final value at a level 118. Each of these transitionscan result in unknown states. With the approximation algorithm that willbe described hereinbelow, it can be seen that, when a transition is madefrom level 110 to level 112, an approximation curve for the dynamicbehavior 120 is provided. When making a transition from level 114 tolevel 116, an approximation gain curve 124 is provided to approximatethe steady state gains between the two levels 114 and 116. For makingthe transition from level 116 to level 118, an approximation gain curve126 for the steady-state gain is provided. It can therefore be seen thatthe approximation curves 120-126 account for transitions betweensteady-state values that are determined by the network, it being notedthat these are approximations which primarily maintain the steady-stategain within some type of error envelope, the envelope 106 in FIG. 7.

[0074] The approximation is provided by the block 41 noted in FIG. 2 andcan be designed upon a number of criteria, depending upon the problemthat it will be utilized to solve. The system in the preferredembodiment, which is only one example, is designed to satisfy thefollowing criteria:

[0075] 1. Computational Complexity: The approximate systematic modelwill be used in a Model Predictive Control algorithm, therefore, it isrequired to have low computational complexity.

[0076] 2. Localized Accuracy: The steady-state model is accurate inlocalized regions. These regions represent the steady-state operatingregimes of the process. The steady-state model is significantly lessaccurate outside these localized regions.

[0077] 3. Final Steady-State: Given a steady-state set point change, anoptimization algorithm which uses the steady-state model will be used tocompute the steady-state inputs required to achieve the set point.Because of item 2, it is assumed that the initial and finalsteady-states associated with a set-point change are located in regionsaccurately modeled by the steady-state model.

[0078] Given the noted criteria, an approximate systematic model can beconstructed by enforcing consistency of the steady-state and dynamicmodel at the initial and final steady-state associated with a set pointchange and utilizing a linear approximation at points in between the twosteady-states. This approximation guarantees that the model is accuratein regions where the steady-state model is well known and utilizes alinear approximation in regions where the steady-state model is known tobe less accurate. In addition, the resulting model has low computationalcomplexity. For purposes of this proof, Equation 13 is modified asfollows: $\begin{matrix}{b_{i,{s\quad c\quad a\quad l\quad e\quad d}} = \frac{b_{i}{K_{s\quad s}\left( {u\left( {t - d - 1} \right)} \right)}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}{\sum\limits_{i = 1}^{n}b_{i}}} & (14)\end{matrix}$

[0079] This new equation 14 utilizes K_(ss)(u(t−d−1)) instead ofK_(ss)(u(t)) as the consistent gain, resulting in a systematic modelwhich is delay invariant.

[0080] The approximate systematic model is based upon utilizing thegains associated with the initial and final steady-state values of aset-point change. The initial steady-state gain is denoted K^(l) _(ss)while the initial steady-state input is given by U^(i) _(ss). The finalsteady-state gain is K^(f) _(ss) and the final input is U^(f) _(ss).Given these values, a linear approximation to the gain is given by:$\begin{matrix}{{K_{s\quad s}\left( {u(t)} \right)} = {K_{ss}^{i} + {\frac{K_{ss}^{f} - K_{ss}^{i}}{U_{ss}^{f} - U_{ss}^{i}}{\left( {{u(t)} - U_{s\quad s}^{i}} \right).}}}} & (15)\end{matrix}$

[0081] Substituting this approximation into Equation 13 and replacingu(t−d−1)−u^(i) by δu(t−d−1) yields: $\begin{matrix}{{\overset{\sim}{b}}_{j,{s\quad c\quad a\quad l\quad e\quad d}} = {\frac{b_{j}{K_{s\quad s}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}} + {\frac{1}{2}\frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{ss}^{f} - K_{ss}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{ss}^{f} - U_{ss}^{i}} \right)}\delta \quad {{u\left( {t - d - i} \right)}.}}}} & (16)\end{matrix}$

[0082] To simplify the expression, define the variable b_(j)-Bar as:$\begin{matrix}{{\overset{\_}{b}}_{j} = \frac{b_{j}{K_{s\quad s}^{i}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}}{\sum\limits_{i = 1}^{n}b_{i}}} & (17)\end{matrix}$

[0083] and g_(j) as: $\begin{matrix}{g_{j} = \frac{{b_{j}\left( {1 + {\sum\limits_{i = 1}^{n}a_{i}}} \right)}\left( {K_{s\quad s}^{f} - K_{s\quad s}^{i}} \right)}{\left( {\sum\limits_{i = 1}^{n}b_{i}} \right)\left( {U_{s\quad s}^{f} - U_{s\quad s}^{i}} \right)}} & (18)\end{matrix}$

[0084] Equation 16 may be written as:

{tilde over (b)} _(j,scaled) ={overscore (b)} _(j) +g _(j)δu(t−d−i).  (19)

[0085] Finally, substituting the scaled b's back into the originaldifference Equation 7, the following expression for the approximatesystematic model is obtained: $\begin{matrix}{{\delta \quad {y(t)}} = {{\sum\limits_{i = 1}^{n}{{\overset{\_}{b}}_{i}\delta \quad {u\left( {t - d - i} \right)}}} + {\sum\limits_{i = 1}^{n}{g_{i}\delta \quad {u\left( {t - d - i^{2}} \right)}\delta \quad {u\left( {t - d - i} \right)}}} - {\sum\limits_{i = 1}^{n}{a_{i}\delta \quad {y\left( {t - i} \right)}}}}} & (20)\end{matrix}$

[0086] The linear approximation for gain results in a quadraticdifference equation for the output. Given Equation 20, the approximatesystematic model is shown to be of low computational complexity. It maybe used in a MPC algorithm to efficiently compute the required controlmoves for a transition from one steady-state to another after aset-point change. Note that this applies to the dynamic gain variationsbetween steady-state transitions and not to the actual path values.

[0087] Control System Error Constraints

[0088] Referring now to FIG. 9, there is illustrated a block diagram ofthe prediction engine for the dynamic controller 82 of FIG. 6. Theprediction engine is operable to essentially predict a value of y(t) asthe predicted future value y(t+1). Since the prediction engine mustdetermine what the value of the output y(t) is at each future valuebetween two steady-state values, it is necessary to perform these in a“step” manner. Therefore, there will be k steps from a value of zero toa value of N, which value at k=N is the value at the “horizon”, thedesired value. This, as will be described hereinbelow, is an iterativeprocess, it being noted that the terminology for “(t+1)” refers to anincremental step, with an incremental step for the dynamic controllerbeing smaller than an incremented step for the steady-state model. Forthe steady-state model, “y(t+N)” for the dynamic model will be, “y(t+1)”for the steady state The value y(t+1) is defined as follows:

y(t+1)=a ₁ y(t)+a ₂ y(t−1)+b ₁ u(t−d−1)+b ₂ u(t−d−2)  (021)

[0089] With further reference to FIG. 9, the input values u(t) for each(u,y) pair are input to a delay line 140. The output of the delay lineprovides the input value u(t) delayed by a delay value “d”. There areprovided only two operations for multiplication with the coefficients b₁and b₂, such that only two values u(t) and u(t−1) are required. Theseare both delayed and then multiplied by the coefficients b₁ and b₂ andthen input to a summing block 141. Similarly, the output value y^(p)(t)is input to a delay line 142, there being two values required formultiplication with the coefficients a₁ and a₂. The output of thismultiplication is then input to the summing block 141. The input to thedelay line 142 is either the actual input value y^(a)(t) or the iteratedoutput value of the summation block 141, which is the previous valuecomputed by the dynamic controller 82. Therefore, the summing block 141will output the predicted value y(t+1) which will then be input to amultiplexor 144. The multiplexor 144 is operable to select the actualoutput y^(a)(t) on the first operation and, thereafter, select theoutput of the summing block 141. Therefore, for a step value of k=0 thevalue y^(a)(t) will be selected by the multiplexor 144 and will belatched in a latch 145. The latch 145 will provide the predicted valuey^(p)(t+k) on an output 146. This is the predicted value of y(t) for agiven k that is input back to the input of delay line 142 formultiplication with the coefficients a₁ and a₂. This is iterated foreach value of k from k=0 to k=N.

[0090] The a₁ and a₂ values are fixed, as described above, with the b₁and b₂ values scaled. This scaling operation is performed by thecoefficient modification block 38. However, this only defines thebeginning steady-state value and the final steady-state value, with thedynamic controller and the optimization routines described in thepresent application defining how the dynamic controller operates betweenthe steady-state values and also what the gain of the dynamic controlleris. The gain specifically is what determines the modification operationperformed by the coefficient modification block 38.

[0091] In FIG. 9, the coefficients in the coefficient modification block38 are modified as described hereinabove with the information that isderived from the steady-state model. The steady-state model is operatedin a control application, and is comprised in part of a forwardsteady-state model 141 which is operable to receive the steady-stateinput value U_(ss)(t) and predict the steady-state output valueY_(ss)(t). This predicted value is utilized in an inverse steady-statemodel 143 to receive the desired value y^(d)(t) and the predicted outputof the steady-state model 141 and predict a future steady-state inputvalue or manipulated value U_(ss)(t+N) and also a future steady-stateinput value Y_(ss)(t+N) in addition to providing the steady-state gainK_(ss). As described hereinabove, these are utilized to generate scaledb-values. These b-values are utilized to define the gain k_(d) of thedynamic model. In can therefore be seen that this essentially takes alinear dynamic model with a fixed gain and allows it to have a gainthereof modified by a non-linear model as the operating point is movedthrough the output space.

[0092] Referring now to FIG. 10, there is illustrated a block diagram ofthe dynamic controller and optimizer. The dynamic controller includes adynamic model 149 which basically defines the predicted value y^(p)(k)as a function of the inputs y(t), s(t) and u(t). This was essentiallythe same model that was described hereinabove with reference to FIG. 9.The model 149 predicts the output values y^(p)(k) between the twosteady-state values, as will be described hereinbelow. The model 149 ispredefined and utilizes an identification algorithm to identify the a₁,a₂, b₁ and b₂ coefficients during training. Once these are identified ina training and identification procedure, these are “fixed”. However, asdescribed hereinabove, the gain of the dynamic model is modified byscaling the coefficients b₁ and b₂. This gain scaling is not describedwith respect to the optimization operation of FIG. 10, although it canbe incorporated in the optimization operation.

[0093] The output of model 149 is input to the negative input of asumming block 150. Summing block 150 sums the predicted output y^(p)(k)with the desired output y^(d)(t). In effect, the desired value ofy^(d)(t) is effectively the desired steady-state value Y^(f) _(ss)although it can be any desired value. The output of the summing block150 comprises an error value which is essentially the difference betweenthe desired value y^(d)(t) and the predicted value y^(p)(k). The errorvalue is modified by an error modification block 151, as will bedescribed hereinbelow, in accordance with error modification parametersin a block 152. The modified error value is then input to an inversemodel 153, which basically performs an optimization routine to predict achange in the input value u(t). In effect, the optimizer 153 is utilizedin conjunction with the model 149 to minimize the error output bysumming block 150. Any optimization function can be utilized, such as aMonte Carlo procedure. However, in the present invention, a gradientcalculation is utilized. In the gradient method, the gradient ∂(y)/∂(u)is calculated and then a gradient solution performed as follows:$\begin{matrix}{{\Delta \quad u_{n\quad e\quad w}} = {{\Delta \quad u_{o\quad l\quad d}} + {\left( \frac{\partial(y)}{\partial(u)} \right) \times E}}} & (022)\end{matrix}$

[0094] The optimization function is performed by the inverse model 153in accordance with optimization constraints in a block 154. An iterationprocedure is performed with an iterate block 155 which is operable toperform an iteration with the combination of the inverse model 153 andthe predictive model 149 and output on an output line 156 the futurevalue u(t+k+1). For k=0, this will be the initial steady-state value andfor k=N, this will be the value at the horizon, or at the nextsteady-state value. During the iteration procedure, the previous valueof u(t+k) has the change value Δu added thereto. This value is utilizedfor that value of k until the error is within the appropriate levels.Once it is at the appropriate level, the next u(t+k) is input to themodel 149 and the value thereof optimized with the iterate block 155.Once the iteration procedure is done, it is latched. As will bedescribed hereinbelow, this is a combination of modifying the error suchthat the actual error output by the block 150 is not utilized by theoptimizer 153 but, rather, a modified error is utilized. Alternatively,different optimization constraints can be utilized, which are generatedby the block 154, these being described hereinbelow.

[0095] Referring now to FIGS. 11a and 11 b, there are illustrated plotsof the output y(t+k) and the input u_(k)(t+k+1), for each k from theinitial steady-state value to the horizon steady-state value at k=N.With specific reference to FIG. 11a, it can be seen that theoptimization procedure is performed utilizing multiple passes. In thefirst pass, the actual value u^(a)(t+k) for each k is utilized todetermine the values of y(t+k) for each u,y pair. This is thenaccumulated and the values processed through the inverse model 153 andthe iterate block 155 to minimize the error. This generates a new set ofinputs u_(k)(t+k+1) illustrated in FIG. 11b. Therefore, the optimizationafter pass 1 generates the values of u(t+k+1) for the second pass. Inthe second pass, the values are again optimized in accordance with thevarious constraints to again generate another set of values foru(t+k+1). This continues until the overall objective function isreached. This objective function is a combination of the operations as afunction of the error and the operations as a function of theconstraints, wherein the optimization constraints may control theoverall operation of the inverse model 153 or the error modificationparameters in block 152 may control the overall operation. Each of theoptimization constraints will be described in more detail hereinbelow.

[0096] Referring now to FIG. 12, there is illustrated a plot of y^(d)(t)and y^(p)(t). The predicted value is represented by a waveform 170 andthe desired output is represented by a waveform 172, both plotted overthe horizon between an initial steady-state value Y^(l) _(ss) and afinal steady-state value Y^(f) _(ss). It can be seen that the desiredwaveform prior to k=0 is substantially equal to the predicted output. Atk=0, the desired output waveform 172 raises its level, thus creating anerror. It can be seen that at k=0, the error is large and the systemthen must adjust the manipulated variables to minimize the error andforce the predicted value to the desired value. The objective functionfor the calculation of error is of the form: $\begin{matrix}{\min\limits_{\Delta \quad u_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {{{\overset{\rightarrow}{y}}^{p}(t)} - {{\overset{\rightarrow}{y}}^{d}(t)}} \right)^{2}} \right.}}} & (23)\end{matrix}$

[0097] where:

[0098] Du_(il) is the change in input variable (IV) I at time interval l

[0099] A_(j) is the weight factor for control variable (CV) j

[0100] y^(p)(t) is the predicted value of CV j at time interval k

[0101] y^(d)(t) is the desired value of CV j.

[0102] Trajectory Weighting

[0103] The present system utilizes what is referred to as “trajectoryweighting” which encompasses the concept that one does not put aconstant degree of importance on the future predicted process behaviormatching the desired behavior at every future time set, i.e., at lowk-values. One approach could be that one is more tolerant of error inthe near term (low k-values) than farther into the future (highk-values). The basis for this logic is that the final desired behavioris more important than the path taken to arrive at the desired behavior,otherwise the path traversed would be a step function. This isillustrated in FIG. 13 wherein three possible predicted behaviors areillustrated, one represented by a curve 174 which is acceptable, one isrepresented by a different curve 176, which is also acceptable and onerepresented by a curve 178, which is unacceptable since it goes abovethe desired level on curve 172. Curves 174-178 define the desiredbehavior over the horizon for k=1 to N.

[0104] In Equation 23, the predicted curves 174-178 would be achieved byforcing the weighting factors A_(j) to be time varying. This isillustrated in FIG. 14. In FIG. 14, the weighting factor A as a functionof time is shown to have an increasing value as time and the value of kincreases. This results in the errors at the beginning of the horizon(low k-values) being weighted much less than the errors at the end ofthe horizon (high k-values). The result is more significant than merelyredistributing the weights out to the end of the control horizon at k=N.This method also adds robustness, or the ability to handle a mismatchbetween the process and the prediction model. Since the largest error isusually experienced at the beginning of the horizon, the largest changesin the independent variables will also occur at this point. If there isa mismatch between the process and the prediction (model error), theseinitial moves will be large and somewhat incorrect, which can cause poorperformance and eventually instability. By utilizing the trajectoryweighting method, the errors at the beginning of the horizon areweighted less, resulting in smaller changes in the independent variablesand, thus, more robustness.

[0105] Error Constraints

[0106] Referring now to FIG. 15, there are illustrated constraints thatcan be placed upon the error. There is illustrated a predicted curve 180and a desired curve 182, desired curve 182 essentially being a flatline. It is desirable for the error between curve 180 and 182 to beminimized. Whenever a transient occurs at t=0, changes of some sort willbe required. It can be seen that prior to t=0, curve 182 and 180 aresubstantially the same, there being very little error between the two.However, after some type of transition, the error will increase. If arigid solution were utilized, the system would immediately respond tothis large error and attempt to reduce it in as short a time aspossible. However, a constraint frustum boundary 184 is provided whichallows the error to be large at t=0 and reduces it to a minimum level ata point 186. At point 186, this is the minimum error, which can be setto zero or to a non-zero value, corresponding to the noise level of theoutput variable to be controlled. This therefore encompasses the sameconcepts as the trajectory weighting method in that final futurebehavior is considered more important that near term behavior. The evershrinking minimum and/or maximum bounds converge from a slack positionat t=0 to the actual final desired behavior at a point 186 in theconstraint frustum method.

[0107] The difference between constraint frustums and trajectoryweighting is that constraint frustums are an absolute limit (hardconstraint) where any behavior satisfying the limit is just asacceptable as any other behavior that also satisfies the limit.Trajectory weighting is a method where differing behaviors havegraduated importance in time. It can be seen that the constraintsprovided by the technique of FIG. 15 requires that the value y^(p)(t) isprevented from exceeding the constraint value. Therefore, if thedifference between y^(d)(t) and y^(p)(t) is greater than that defined bythe constraint boundary, then the optimization routine will force theinput values to a value that will result in the error being less thanthe constraint value. In effect, this is a “clamp” on the differencebetween y^(p)(t) and y^(d)(t). In the trajectory weighting method, thereis no “clamp” on the difference therebetween; rather, there is merely anattenuation factor placed on the error before input to the optimizationnetwork.

[0108] Trajectory weighting can be compared with other methods, therebeing two methods that will be described herein, the dynamic matrixcontrol (DMC) algorithm and the identification and command (IdCom)algorithm. The DMC algorithm utilizes an optimization to solve thecontrol problem by minimizing the objective function: $\begin{matrix}{\min\limits_{\Delta \quad U_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {{A_{j}*\left( {{{\overset{\rightarrow}{y}}^{P}(t)} - {{\overset{\rightarrow}{y}}^{D}(t)}} \right)} + {\sum\limits_{i}{B_{i}*{\sum\limits_{1}\left( {\Delta \quad U_{il}} \right)^{2}}}}} \right.}}} & (24)\end{matrix}$

[0109] where B_(i) is the move suppression factor for input variable I.This is described in Cutler, C. R. and B. L. Ramaker, Dynamic MatrixControl—A Computer Control Algorithm, AIChE National Meeting, Houston,Tex. (April, 1979), which is incorporated herein by reference.

[0110] It is noted that the weights A_(j) and desired values y^(d)(t)are constant for each of the control variables. As can be seen fromEquation 24, the optimization is a trade off between minimizing errorsbetween the control variables and their desired values and minimizingthe changes in the independent variables. Without the move suppressionterm, the independent variable changes resulting from the set pointchanges would be quite large due to the sudden and immediate errorbetween the predicted and desired values. Move suppression limits theindependent variable changes, but for all circumstances, not just theinitial errors.

[0111] The IdCom algorithm utilizes a different approach. Instead of aconstant desired value, a path is defined for the control variables totake from the current value to the desired value. This is illustrated inFIG. 16. This path is a more gradual transition from one operation pointto the next. Nevertheless, it is still a rigidly defined path that mustbe met. The objective function for this algorithm takes the form:$\begin{matrix}{\min\limits_{\Delta \quad U_{il}}{\sum\limits_{j}{\sum\limits_{k}\left( {A_{j}*\left( {Y^{P_{j\quad k}} - y_{refjk}} \right)} \right)^{2}}}} & (25)\end{matrix}$

[0112] This technique is described in Richalet, J., A. Rault, J. L.Testud, and J. Papon, Model Predictive Heuristic Control: Applicationsto Industrial Processes, Automatica, 14, 413-428 (1978), which isincorporated herein by reference. It should be noted that therequirement of Equation 25 at each time interval is sometimes difficult.In fact, for control variables that behave similarly, this can result inquite erratic independent variable changes due to the control algorithmattempting to endlessly meet the desired path exactly.

[0113] Control algorithms such as the DMC algorithm that utilize a formof matrix inversion in the control calculation, cannot handle controlvariable hard constraints directly. They must treat them separately,usually in the form of a steady-state linear program. Because this isdone as a steady-state problem, the constraints are time invariant bydefinition. Moreover, since the constraints are not part of a controlcalculation, there is no protection against the controller violating thehard constraints in the transient while satisfying them at steady-state.

[0114] With further reference to FIG. 15, the boundaries at the end ofthe envelope can be defined as described hereinbelow. One techniquedescribed in the prior art, W. Edwards Deming, “Out of the Crisis,”Massachusetts Institute of Technology, Center for Advanced EngineeringStudy, Cambridge Mass., Fifth Printing, September 1988, pages 327-329,describes various Monte Carlo experiments that set forth the premisethat any control actions taken to correct for common process variationactually may have a negative impact, which action may work to increasevariability rather than the desired effect of reducing variation of thecontrolled processes. Given that any process has an inherent accuracy,there should be no basis to make a change based on a difference thatlies within the accuracy limits of the system utilized to control it. Atpresent, commercial controllers fail to recognize the fact that changesare undesirable, and continually adjust the process, treating alldeviation from target, no matter how small, as a special cause deservingof control actions, i.e., they respond to even minimal changes. Overadjustment of the manipulated variables therefore will result, andincrease undesirable process variation. By placing limits on the errorwith the present filtering algorithms described herein, only controlleractions that are proven to be necessary are allowed, and thus, theprocess can settle into a reduced variation free from unmeritedcontroller disturbances. The following discussion will deal with onetechnique for doing this, this being based on statistical parameters.

[0115] Filters can be created that prevent model-based controllers fromtaking any action in the case where the difference between thecontrolled variable measurement and the desired target value are notsignificant. The significance level is defined by the accuracy of themodel upon which the controller is statistically based. This accuracy isdetermined as a function of the standard deviation of the error and apredetermined confidence level. The confidence level is based upon theaccuracy of the training. Since most training sets for a neuralnetwork-based model will have “holes” therein, this will result ininaccuracies within the mapped space. Since a neural network is anempirical model, it is only as accurate as the training data set. Eventhough the model may not have been trained upon a given set of inputs,it will extrapolate the output and predict a value given a set ofinputs, even though these inputs are mapped across a space that isquestionable. In these areas, the confidence level in the predictedoutput is relatively low. This is described in detail in U.S. patentapplication Ser. No. 08/025,184, filed Mar. 2, 1993, which isincorporated herein by reference.

[0116] Referring now to FIG. 17, there is illustrated a flowchartdepicting the statistical method for generating the filter and definingthe end point 186 in FIG. 15. The flowchart is initiated at a startblock 200 and then proceeds to a function block 202, wherein the controlvalues u(t+1) are calculated. However, prior to acquiring these controlvalues, the filtering operation must be a processed. The program willflow to a function block 204 to determine the accuracy of thecontroller. This is done off-line by analyzing the model predictedvalues compared to the actual values, and calculating the standarddeviation of the error in areas where the target is undisturbed. Themodel accuracy of e_(m)(t) is defined as follows:

e _(m)(t)=a(t)−p(t)  (026)

[0117] where:

[0118] e_(m)=model error,

[0119] a=actual value

[0120] p=model predicted value

[0121] The model accuracy is defined by the following equation:

Acc=H*σ _(m)  (027)

[0122] where:

[0123] Acc=accuracy in terms of minimal detector error

[0124] H=significance level=1 67% confidence

[0125] =2 95% confidence

[0126] =3 99.5% confidence

[0127] σ_(m)=standard deviation of e_(m)(t).

[0128] The program then flows to a function block 206 to compare thecontroller error e_(c)(t) with the model accuracy. This is done bytaking the difference between the predicted value (measured value) andthe desired value. This is the controller error calculation as follows:

e _(c)(t)=d(t)−m(t)  (028)

[0129] where:

[0130] e_(c)=controller error

[0131] d=desired value

[0132] m=measured value

[0133] The program will then flow to a decision block 208 to determineif the error is within the accuracy limits. The determination as towhether the error is within the accuracy limits is done utilizingShewhart limits. With this type of limit and this type of filter, adetermination is made as to whether the controller error e_(c)(t) meetsthe following conditions: e_(c)(t)≧−1*Acc and e_(c)(t)≦+1*Acc, theneither the control action is suppressed or not suppressed. If it iswithin the accuracy limits, then the control action is suppressed andthe program flows along a “Y” path. If not, the program will flow alongthe “N” path to function block 210 to accept the u(t+1) values. If theerror lies within the controller accuracy, then the program flows alongthe “Y” path from decision block 208 to a function block 212 tocalculate the running accumulation of errors. This is formed utilizing aCUSUM approach. The controller CUSUM calculations are done as follows:

S _(low)=min(0, S _(low)(t−1)+d(t)−m(t))−Σ(m)+k)  (029)

S _(hi)=max(0, S _(hi)(t−1)+[d(t)−m(t))−Σ(m)]−k)  (030)

[0134] where:

[0135] S_(hi)=Running Positive Qsum

[0136] S_(low) 32 Running Negative Qsum

[0137] k=Tuning factor−minimal detectable change threshold with thefollowing defined:

[0138] Hq=significance level. Values of (j,k) can be found so that theCUSUM control chart will have significance levels equivalent to Shewhartcontrol charts.

[0139] The program will then flow to a decision block 214 to determineif the CUSUM limits check out, i.e., it will determine if the Qsumvalues are within the limits. If the Qsum, the accumulated sum error, iswithin the established limits, the program will then flow along the “Y”path. And, if it is not within the limits, it will flow along the “N”path to accept the controller values u(t+1). The limits are determinedif both the value of S_(hi)≧+1*Hq and S_(low)≦−1*Hq. Both of theseactions will result in this program flowing along the “Y” path. If itflows along the “N” path, the sum is set equal to zero and then theprogram flows to the function block 210. If the Qsum values are withinthe limits, it flows along the “Y” path to a function block 218 whereina determination is made as to whether the user wishes to perturb theprocess. If so, the program will flow along the “Y” path to the functionblock 210 to accept the control values u(t+1). If not, the program willflow along the “N” path from decision block 218 to a function block 222to suppress the controller values u(t+1). The decision block 218, whenit flows along the “Y” path, is a process that allows the user tore-identify the model for on-line adaptation, i.e., retrain the model.This is for the purpose of data collection and once the data has beencollected, the system is then reactivated.

[0140] Referring now to FIG. 18, there is illustrated a block diagram ofthe overall optimization procedure. In the first step of the procedure,the initial steady-state values {Y_(ss) ^(i), U_(ss) ^(i)} and the finalsteady-state values {Y_(ss) ^(f), U_(ss) ^(f)} are determined, asdefined in blocks 226 and 228, respectively. In some calculations, boththe initial and the final steady-state values are required. The initialsteady-state values are utilized to define the coefficients a^(l), b^(l)in a block 228. As described above, this utilizes the coefficientscaling of the b-coefficients. Similarly, the steady-state values inblock 228 are utilized to define the coefficients a^(f), b^(f), it beingnoted that only the b-coefficients are also defined in a block 229. Oncethe beginning and end points are defined, it is then necessary todetermine the path therebetween. This is provided by block 230 for pathoptimization. There are two methods for determining how the dynamiccontroller traverses this path. The first, as described above, is todefine the approximate dynamic gain over the path from the initial gainto the final gain. As noted above, this can incur some instabilities.The second method is to define the input values over the horizon fromthe initial value to the final value such that the desired value Y_(ss)^(f) is achieved. Thereafter, the gain can be set for the dynamic modelby scaling the b-coefficients. As noted above, this second method doesnot necessarily force the predicted value of the output y^(p)(t) along adefined path; rather, it defines the characteristics of the model as afunction of the error between the predicted and actual values over thehorizon from the initial value to the final or desired value. Thiseffectively defines the input values for each point on the trajectoryor, alternatively, the dynamic gain along the trajectory.

[0141] Referring now to FIG. 18a, there is illustrated a diagrammaticrepresentation of the manner in which the path is mapped through theinput and output space. The steady-state model is operable to predictboth the output steady-state value Y_(ss) ^(l) at a value of k=0, theinitial steady-state value, and the output steady-state value Y_(ss)^(i) at a time t+N where k=N, the final steady-state value. At theinitial steady-state value, there is defined a region 227, which region227 comprises a surface in the output space in the proximity of theinitial steady-state value, which initial steady-state value also liesin the output space. This defines the range over which the dynamiccontroller can operate and the range over which it is valid. At thefinal steady-state value, if the gain were not changed, the dynamicmodel would not be valid. However, by utilizing the steady-state modelto calculate the steady-state gain at the final steady-state value andthen force the gain of the dynamic model to equal that of thesteady-state model, the dynamic model then becomes valid over a region229, proximate the final steady-state value. This is at a value of k=N.The problem that arises is how to define the path between the initialand final steady-state values. One possibility, as mentionedhereinabove, is to utilize the steady-state model to calculate thesteady-state gain at multiple points along the path between the initialsteady-state value and the final steady-state value and then define thedynamic gain at those points. This could be utilized in an optimizationroutine, which could require a large number of calculations. If thecomputational ability were there, this would provide a continuouscalculation for the dynamic gain along the path traversed between theinitial steady-state value and the final steady-state value utilizingthe steady-state gain. However, it is possible that the steady-statemodel is not valid in regions between the initial and final steady-statevalues, i.e., there is a low confidence level due to the fact that thetraining in those regions may not be adequate to define the modeltherein. Therefore, the dynamic gain is approximated in these regions,the primary goal being to have some adjustment of the dynamic modelalong the path between the initial and the final steady-state valuesduring the optimization procedure. This allows the dynamic operation ofthe model to be defined. This is represented by a number of surfaces 225as shown in phantom.

[0142] Referring now to FIG. 19, there is illustrated a flow chartdepicting the optimization algorithm. The program is initiated at astart block 232 and then proceeds to a function block 234 to define theactual input values u^(a)(t) at the beginning of the horizon, thistypically being the steady-state value U_(ss). The program then flows toa function block 235 to generate the predicted values y^(p)(k) over thehorizon for all k for the fixed input values. The program then flows toa function block 236 to generate the error E(k) over the horizon for allk for the previously generated y^(p)(k). These errors and the predictedvalues are then accumulated, as noted by function block 238. The programthen flows to a function block 240 to optimize the value of u(t) foreach value of k in one embodiment. This will result in k-values foru(t). Of course, it is sufficient to utilize less calculations than thetotal k-calculations over the horizon to provide for a more efficientalgorithm. The results of this optimization will provide the predictedchange Δu(t+k) for each value of k in a function block 242. The programthen flows to a function block 243 wherein the value of u(t+k) for eachu will be incremented by the value Δu(t+k). The program will then flowto a decision block 244 to determine if the objective function notedabove is less than or equal to a desired value. If not, the program willflow back along an “N” path to the input of function block 235 to againmake another pass. This operation was described above with respect toFIGS. 11a and 11 b. When the objective function is in an acceptablelevel, the program will flow from decision block 244 along the “Y” pathto a function block 245 to set the value of u(t+k) for all u. Thisdefines the path. The program then flows to an End block 246.

[0143] Steady State Gain Determination

[0144] Referring now to FIG. 20, there is illustrated a plot of theinput space and the error associated therewith. The input space iscomprised of two variables x₁ and x₂. The y-axis represents the functionf(x₁, x₂). In the plane of x₁ and x₂, there is illustrated a region 250,which represents the training data set. Areas outside of the region 250constitute regions of no data, i.e., a low confidence level region. Thefunction Y will have an error associated therewith. This is representedby a plane 252. However, the error in the plane 250 is only valid in aregion 254, which corresponds to the region 250. Areas outside of region254 on plane 252 have an unknown error associated therewith. As aresult, whenever the network is operated outside of the region 250 withthe error region 254, the confidence level in the network is low. Ofcourse, the confidence level will not abruptly change once outside ofthe known data regions but, rather, decreases as the distance from theknown data in the training set increases. This is represented in FIG. 21wherein the confidence is defined as α(x). It can be seen from FIG. 21that the confidence level α(x) is high in regions overlying the region250.

[0145] Once the system is operating outside of the training dataregions, i.e., in a low confidence region, the accuracy of the neuralnet is relatively low. In accordance with one aspect of the preferredembodiment, a first principles model g(x) is utilized to governsteady-state operation. The switching between the neural network modelf(x) and the first principle models g(x) is not an abrupt switching but,rather, it is a mixture of the two.

[0146] The steady-state gain relationship is defined in Equation 7 andis set forth in a more simple manner as follows: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {f\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (031)\end{matrix}$

[0147] A new output function Y(u) is defined to take into account theconfidence factor α(u) as follows:

Y({right arrow over (u)})=α({right arrow over (u)}).f({right arrow over(u)})+(1−α({right arrow over (u)}))g({right arrow over (u)})  (032)

[0148] where:

[0149] α(u)=confidence in model f(u)

[0150] α(u) in the range of 0→1

[0151] α(u)∈{0,1}

[0152] This will give rise to the relationship: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = \frac{\partial\left( {Y\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} & (033)\end{matrix}$

[0153] In calculating the steady-state gain in accordance with thisEquation utilizing the output relationship Y(u), the following willresult: $\begin{matrix}{{K\left( \overset{\rightarrow}{u} \right)} = {{\frac{\partial\left( {\alpha \left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {F\left( \overset{\rightarrow}{u} \right)}} + {{\alpha \left( \overset{\rightarrow}{u} \right)}\frac{\partial\left( {F\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}} + {\frac{\partial\left( {1 - {\alpha \left( \overset{\rightarrow}{u} \right)}} \right)}{\partial\left( \overset{\rightarrow}{u} \right)} \times {g\left( \overset{\rightarrow}{u} \right)}} + {\left( {1 - {\alpha \left( \overset{\rightarrow}{u} \right)}} \right)\frac{\partial\left( {g\left( \overset{\rightarrow}{u} \right)} \right)}{\partial\left( \overset{\rightarrow}{u} \right)}}}} & (034)\end{matrix}$

[0154] Referring now to FIG. 22, there is illustrated a block diagram ofthe embodiment for realizing the switching between the neural networkmodel and the first principles model. A neural network block 300 isprovided for the function f(u), a first principle block 302 is providedfor the function g(u) and a confidence level block 304 for the functionα(u). The input u(t) is input to each of the blocks 300-304. The outputof block 304 is processed through a subtraction block 306 to generatethe function 1−α(u), which is input to a multiplication block 308 formultiplication with the output of the first principles block 302. Thisprovides the function (1−α(u))*g(u). Additionally, the output of theconfidence block 304 is input to a multiplication block 310 formultiplication with the output of the neural network block 300. Thisprovides the function f(u)*α(u). The output of block 308 and the outputof block 310 are input to a summation block 312 to provide the outputY(u).

[0155] Referring now to FIG. 23, there is illustrated an alternateembodiment which utilizes discreet switching. The output of the firstprinciples block 302 and the neural network block 300 are provided andare operable to receive the input x(t). The output of the network block300 and first principles block 302 are input to a switch 320, the switch320 operable to select either the output of the first principals block302 or the output of the neural network block 300. The output of theswitch 320 provides the output Y(u).

[0156] The switch 320 is controlled by a domain analyzer 322. The domainanalyzer 322 is operable to receive the input x(t) and determine whetherthe domain is one that is within a valid region of the network 300. Ifnot, the switch 320 is controlled to utilize the first principlesoperation in the first principles block 302. The domain analyzer 322utilizes the training database 326 to determine the regions in which thetraining data is valid for the network 300. Alternatively, the domainanalyzer 320 could utilize the confidence factor α(u) and compare thiswith a threshold, below which the first principles model 302 would beutilized.

[0157] Although the preferred embodiment has been described in detail,it should be understood that various changes, substitutions andalterations can be made therein without departing from the spirit andscope of the invention as defined by the appended claims.

What is claimed is:
 1. A dynamic controller for controlling theoperation of the plant by predicting a change in the dynamic inputvalues to the plant to effect a change in the output from a currentoutput value at a first time to a desired output value at a second time,comprising: a dynamic predictive model for receiving the current inputvalue and the desired output value and predicting a plurality of inputvalues at different time positions between the first time and the secondtime to define a dynamic operation path of the plant between the currentoutput value and the desired output value at the second time; and anoptimizer for optimizing the operation of the dynamic controller at eachof the different time positions from the first time to the second timein accordance with a predetermined optimization method that optimizesthe objectives of the dynamic controller to achieve a desired path, suchthat the objectives of the dynamic predictive model varies as a functionof time.
 2. The dynamic controller of claim 1, wherein said dynamicpredictive model comprises: a dynamic forward model operable to receiveinput values at each of said time positions and map said received inputvalues through a stored representation of the plant to provide apredicted dynamic output value; an error generator for comparing thepredicted dynamic output value to the desired output value andgenerating a primary error value as the difference therebetween for eachof said time positions; an error minimization device for determining achange in the input value to minimize the primary error value output bysaid error generator; a summation device for summing said determinedinput change value with the original input value for each time positionto provide a future input value; and a controller for controlling theoperation of said error minimization device to operate under control ofsaid optimizer to minimize said primary error value in accordance withsaid predetermined optimization method.
 3. The dynamic controller ofclaim 2, wherein said controller controls the operation of saidsummation device to iteratively minimize said primary error value bystoring the summed output from said summation device in a latch in afirst pass through said error minimization device and input the latchcontents to said dynamic forward model in subsequent pass and for aplurality of subsequent passes, with the output of said errorminimization device summed with the previous contents of said latch withsaid summation device, said latch containing the current value of theinput on the first pass through said dynamic forward model and saiderror minimization device, said controller outputting the contents ofsaid latch as the input to the plant after said primary error value hasbeen determined to meet the objectives in accordance with saidpredetermined optimization method.
 4. The dynamic controller of claim 2,wherein said dynamic forward model is a dynamic linear model with afixed gain.
 5. The dynamic controller of claim 4 and further comprisinga gain adjustment device for adjusting the gain of said linear model forsubstantially all of said time positions.
 6. The dynamic controller ofclaim 5, wherein said gain adjustment device comprises: a non-linearmodel for receiving an input value and mapping the received input valuethrough a stored representation of the plant to provide on the outputthereof a predicted output value, and having a non-linear gainassociated therewith; said linear model having parameters associatedtherewith that define the dynamic gain thereof; and a parameteradjustment device for adjusting the parameters of said linear model as afunction of the gain of said non-linear model for at least one of saidtime positions.
 7. The dynamic controller of claim 6, wherein said gainadjustment device further comprises an approximation device forapproximating the dynamic gain for a plurality of said time positionsbetween the value of the dynamic gain at said first time and thedetermined dynamic gain at the one of said time positions having thedynamic gain thereof determined by said parameter adjustment device. 8.The dynamic controller of claim 7, wherein the one of said timepositions at which said parameter adjustment device adjusts saidparameters as a function of the gain of said non-linear modelcorresponds to the maximum at the second time.
 9. The dynamic controllerof claim 6, wherein said non-linear model is a steady-state model. 10.The dynamic controller of claim 2, wherein said error minimizationdevice includes a primary error modification device for modifying saidprimary error to provide a modified error value, said error minimizationdevice optimizing the operation of the dynamic controller to minimizesaid modified error value in accordance with said predeterminedoptimization method.
 11. The dynamic controller of claim 10, whereinsaid primary error is weighted as a function of time from the first timeto the second time.
 12. The dynamic controller of claim 11, wherein saidweighting function decreases as a function of time such that saidprimary error value is attenuated at a relatively high value proximateto the first time and attenuated at a relatively low level proximate tothe second time.
 13. The dynamic controller of claim 2, wherein saiderror minimization device receives said predicted output from saiddynamic forward model and determines a change in the input valuemaintaining a constraint on the predicted output value such thatminimization of the primary error value through a determined inputchange would not cause said predicted output from said dynamic forwardmodel to exceed said constraint.
 14. The dynamic controller of claim 2,and further comprising a filter determining the operation of said errorminimization device when the difference between the predictedmanipulated variable and the desired output value is insignificant. 15.The dynamic controller of claim 14, wherein said filter determines whenthe difference between the predicted manipulated variable and thedesired output value is not significant by determining the accuracy ofthe model upon which the dynamic forward model is based.
 16. The dynamiccontroller of claim 15, wherein the accuracy is determined as a functionof the standard deviation of the error and a predetermined confidencelevel, wherein said confidence level is based upon the accuracy of thetraining over the mapped space.
 17. A method for predicting an outputvalue from a received input value, comprising the steps of: modeling aset of static data received from a plant in a predictive static modelover a first range, the static model having a static gain of K andmodeling the static operation of the plant; modeling a set of dynamicdata received from the plant in a predictive dynamic model over a secondrange smaller than the first range, the dynamic model having a dynamicgain k and modeling the dynamic operation of the plant, and the dynamicmodel being independent of the operation of the static model; adjustingthe gain of the dynamic model as a predetermined function of the gain ofthe static model to vary the model parameters of the dynamic model;predicting the dynamic operation of the predicted input value for achange in the input value between a first input value at a first timeand a second input value at a second time; subtracting the input valuefrom a steady-state input value previously determined and inputting thedifference to the dynamic model and processing the input through thedynamic model to provide a dynamic output value; and adding the dynamicoutput value from the dynamic model to a steady-state output valuepreviously determined to provide a predicted value.
 18. The method ofclaim 17, wherein the predetermined function is an equality functionwherein the static gain K is equal to the dynamic gain k.
 19. The methodof claim 17, wherein the static model is a non-linear model.
 20. Themethod of claim 19, wherein the dynamic model for a given dynamic gainis linear.
 21. The method of claim 20, wherein the step of adjusting thegain of the dynamic model as a function of the predetermined function ofthe gain of the static model is a non-linear operation.
 22. The methodof claim 17, wherein the static model and the dynamic model are utilizedin a control function to receive as inputs the manipulated inputs of theplant, the actual output from the plant in addition to a desired outputvalue at which the plant is to operate, and then perform the step ofpredicting future manipulated inputs required to achieve the desiredoutput.
 23. The method of claim 22, and further comprising the step ofoptimizing the operation of the dynamic model in accordance with apredetermined optimization method between an initial steady-state valueand a predicted final steady-state value that constitutes an inputcontrol variable to the plant during the control operation.
 24. Themethod of claim 23, wherein the step of optimizing comprises determiningthe dynamic gain k for multiple positions between the input steady-stateinput value and the final predicted steady-state input value thatcomprises the input control values to the plant.
 25. The method of claim24, wherein the step of determining utilizes an algorithm that estimatesthe dynamic gain k independent of the operation of the static model. 26.The method of claim 25, wherein the algorithm is a quadratic equation.27. The method of claim 23, wherein the step of predicting with thedynamic model utilized as a dynamic controller comprises the steps of:predicting the dynamic operation of the plant from the initialsteady-state input value to the predicted steady-state input value toprovide a predicted dynamic operation; comparing the predicted dynamicoperation to the desired steady-state value at the final desired outputvalue and generating an error as the difference therebetween;determining a change in the input value for input to step of predictingthe dynamic operation which will vary the input value thereto; andvarying the change in the input value to minimize the error.
 28. Themethod of claim 27, wherein the step of determining the error comprisesmultiplying the determined error value by a predetermined weightingvalue that is a function of time from the first time to the second time.29. The method of claim 27, wherein the predetermined weighting functionattenuates the error for values proximate in time to the initialsteady-state value at the first time and decreases the attenuation valueas time increases to the final steady-state value at the second time.30. A predictive system for predicting the operation of a plant, thepredictive system operable to receive an input value and provide on anoutput of the predictive system a predicted output value, comprising: anon-linear model having an input for receiving the input value andmapping it across a stored learned representation of the plant toprovide the predictive system output on an output, said non-linear modelhaving an internal integrity that is a function of a training operationthat varies across the mapped space such that the accuracy of thepredicted value will vary as the integrity varies; a first principlesmodel for providing a calculated representation of the plant that is nota function of a training operation; a domain analyzer for determiningwhen the input value input to said non-linear model falls within aregion of the mapped space having an integrity that is less than apredetermined integrity threshold; and a control system for selectingbetween said non-linear model and said first principles model based uponsaid domain analyzer determining that the integrity is above or belowthe predetermined threshold, such that said non-linear model is selectedwhen said integrity is above said threshold and said first principlesmodel is selected when said integrity is below said threshold.